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Mathematics > Combinatorics

Title:Hypergraph limits: a regularity approach

Abstract: A sequence of $k$-uniform hypergraphs $H_1, H_2, \dots$ is convergent if the
sequence of homomorphism densities $t(F, H_1), t(F, H_2), \dots$ converges for
every $k$-uniform hypergraph $F$. For graphs, Lovász and Szegedy showed that
every convergent sequence has a limit in the form of a symmetric measurable
function $W \colon [0,1]^2 \to [0,1]$. For hypergraphs, analogous limits $W
\colon [0,1]^{2^k-2} \to [0,1]$ were constructed by Elek and Szegedy using
ultraproducts. These limits had also been studied earlier by Hoover, Aldous,
and Kallenberg in the setting of exchangeable random arrays.
In this paper, we give a new proof and construction of hypergraph limits. Our
approach is inspired by the original approach of Lovász and Szegedy, with the
key ingredient being a weak Frieze-Kannan type regularity lemma.